## Vqc

1 CA

a—/ n2T-1/2 exp(-hv/kT) dr 4n Jo where we assume g = constant and adopt a pure hydrogen plasma, so Z = 1 and ni = ne = n. The integration is along the line of sight through the entire depth A of the cloud; recall that the cloud is optically thin. If one further assumes that the densities and temperature are constant through the depth of the cloud,

This function is sketched in Fig. 3 for two temperatures, T2 > T1, on linear, semilog and log-log axes. The unspecified proportionality constants in (17) are taken to be equal in the two cases; thus Fig. 3 compares the spectra of two hydrogen plasmas that have different temperatures, but which are otherwise identical.

The linear plot (Fig. 3a) shows the typical exponential decrease with increasing frequency. The higher temperature for the T2 curve is reflected in two ways. It has less amplitude at v = 0, as a consequence of the T-1/2 dependence, and it falls to 1/e of its v = 0 amplitude at a greater frequency than does the T1 curve. The higher temperature plasma emits more high-energy photons as might be expected.

Figure 11.3. Sketches of exponential spectra on linear-linear, semilog, and log-log plots for two sources with differing temperatures, T2 and T1. In each case, most of the energy is emitted at the higher frequencies. Measurements of the flux at points C and D permit one to solve for the temperature T of the plasma as well as the factor ne2A where ne is the electron density and A is the line-of-sight thickness of the cloud. The dashed line in (c) illustrates the effect of the Gaunt factor for the T2 curve in a realistic thermal bremsstrahlung spectrum. These plots do not show the prolific emission lines typical of a real plasma.

Figure 11.3. Sketches of exponential spectra on linear-linear, semilog, and log-log plots for two sources with differing temperatures, T2 and T1. In each case, most of the energy is emitted at the higher frequencies. Measurements of the flux at points C and D permit one to solve for the temperature T of the plasma as well as the factor ne2A where ne is the electron density and A is the line-of-sight thickness of the cloud. The dashed line in (c) illustrates the effect of the Gaunt factor for the T2 curve in a realistic thermal bremsstrahlung spectrum. These plots do not show the prolific emission lines typical of a real plasma.

The semilog plot (Fig. 3b) shows the same features with the typical straight-line character of exponential functions.

The log-log plot (Fig. 3c) is interesting; it shows a flat spectrum out to a certain region where it starts to decrease rapidly. (We show the effect of a hypothetical Gaunt factor g with a dashed line.) The turnover occurs near the frequency given by h v ~ kT. At this frequency, the kinetic energy of the emitting electrons, ~ kT, is about equal to the energy of the individual emitted photons h v. Most of the power is emitted in this frequency region. At lower frequencies, hv ^ kT, the exponential function in (17) is approximately unity; recall that an exponential with a small argument (x ^ 1) may be approximated as ex = 1 + x ... ~ 1.0.

On a logarithmic abscissa, the low-frequency portion of the function near hv ^ kT may be stretched out indefinitely toward v = 0, to 0.1 Hz, to 0.01 Hz, to 0.001 Hz, etc.; this is why it is flat. Relatively little power is included in the low-frequency regions because the bandwidths are so small. For example, the band 0.1 to 1 Hz has a bandwidth of 0.9 Hz while the band 0.001-0.01 Hz has a bandwidth of only 0.009 Hz. The power in each decade is proportional to the product of I and the bandwidth. At low frequencies, I is constant but bandwidth becomes negligibly small. This explains qualitatively our statement that most of the emitted power is at frequencies near the cutoff at hv/kT ~ 1.

Measurement of the specific intensity curve I(v) provides direct information about both the product n2A and the temperature T of the cloud. Consider the approximate expression for I(v) (19), for a hydrogen cloud of constant temperature and density along the line of sight with a fixed Gaunt factor. The temperature T is obtained directly from the value of v where I is at e-1 of the maximum, namely at point D in Fig. 3c; at this frequency, hv/kT = 1. The value of I at low frequencies (point C) yields the product n2AT-1/2. Since T is known, we obtain n2A. The unspecified proportionality constant in (19) is a combination of well known physical and numerical constants, so absolute values of Tand n2A are obtained.

Shocks in supernova remnants, stellar coronae, HII regions

The actual spectral form for thermal bremsstrahlung is not a pure exponential. The Gaunt factor causes the flat portion of the log-log plot to decrease slowly with increasing frequency. Also the several atomic elements in cosmic plasmas lead to strong emission lines superposed on the quasi exponential continuum. A theoretical calculation (Fig. 4) of the expected radiation from a plasma of temperature 107 K containing "cosmic" abundances of the elements (e.g. Table 10.2) shows both of these effects.

On the semilog plot of Fig. 4, the Gaunt factor appears at low photon energies hv as an excess above the extrapolated straight-line continuum seen at higher frequencies. This continuum decreases by a factor of about e = 2.7 for each increase of hv by a factor kT as expected from (17). Note that the emission lines typically exceed the continuum in intensity by two decades, a factor of ~100.

Such hot x-ray emitting plasmas are found in shock waves propagating outward from the sites of supernova explosions, for example in the supernova remnant Puppis A (Fig. 5a). Coronae in the vicinity of stars like our sun are hot plasmas of temperatures reaching 106 K and more. In some cases, e.g. Capella (Fig. 5b), the radiation is sufficiently hot and intense that the currently orbiting Chandra observatory can resolve its spectral lines. In both cases, the emission lines are plotted on a linear scale which dramatically illustrates how the spectral line intensities greatly exceed the continuum, in accord with Fig. 4.

Thermal bremsstrahlung emission is found in radio emitting low-temperature plasmas of emission nebulae such as HII regions of the Orion nebula (Fig. 6b). The ideal spectrum for such a source is sketched in Fig. 6a. The flat (a ~ 0) portion is the low-frequency end of an optically thin thermal bremsstrahlung spectrum with g = constant (compare to Fig. 3c).

At very, very low frequencies, plasmas become optically thick; the nebula becomes opaque to its own radiation because the number ofphase space states the photons can occupy are limited at the low frequencies. (Phase space is six-dimensional

hv kT

Figure 11.4. Theoretical x-ray spectrum of an optically thin plasma at T = 107 K on a semilog plot. The ordinate is the log of the volume emissivity j (17) divided by electron density squared. The abscissa is unity at the frequency where the exponential term in (17) equals e-1. The various atomic levels are properly calculated, and strong emission lines are the result. The dashed lines show the effect of photoelectric absorption of x-rays by interstellar gas. [From Tucker and Gould, ApJ 144,244 (1966)]

hv kT

Figure 11.4. Theoretical x-ray spectrum of an optically thin plasma at T = 107 K on a semilog plot. The ordinate is the log of the volume emissivity j (17) divided by electron density squared. The abscissa is unity at the frequency where the exponential term in (17) equals e-1. The various atomic levels are properly calculated, and strong emission lines are the result. The dashed lines show the effect of photoelectric absorption of x-rays by interstellar gas. [From Tucker and Gould, ApJ 144,244 (1966)]

with 3 momentum and 3 position coordinates.) The spectrum thus descends like the low-frequency part of a blackbody spectrum which is similarly constrained by available phase space states. This portion of the spectrum has the v2 Rayleigh-Jeans dependence (see below).

### Synchrotron radiation

A mechanism that can give rise to a spectrum of very different shape is synchrotron radiation. This occurs when very high energy (relativistic) electrons spiral around magnetic field lines due to the q v x B force on an electric charge q moving with velocity v in a magnetic field B. The electrons emit electromagnetic radiation because the spiraling motion constitutes an acceleration, and accelerating charges emit photons. Because the charges are relativistic, the radiation is particularly intense, and it can reach to extremely high energies, even to x rays and gamma rays. It is also

Wavelength ( nm )

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